# Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mapping

• O. O. Shugailo School of Mathematics and Computer Sciences V.N. Karazin Kharkiv National University Kharkiv, Ukraine
Keywords: affine immersion; flat connection; equiaffine structure; Weingarten mapping

### Abstract

In the paper we study equiaffine immersions $f\colon (M^n,\nabla) \rightarrow {\mathbb{R}}^{n+2}$ with flat connection $\nabla$ and one-dimensional Weingarten mapping. For such immersions there are two types of the transversal distribution equiaffine frame.
We give a parametrization of a submanifold with the given properties for both types of equiaffine frame. The main result of the paper is contained in Theorems 1, 2 and Corollary 1: Let $f\colon ({M}^n,\nabla)\rightarrow ({\mathbb{R}}^{n+2},D)$ be an affine immersion with pointwise codimension 2, equiaffine structure, flat connection $\nabla$, one-dimensional Weingarten mapping then there exists three types of its parametrization:
$(i)$
$\vec{r}=g(u^1,\ldots,u^n) \vec{a}_1+\int \vec{\varphi}(u^1)du^1+\sum\limits_{i=2}^n u^i\vec{a}_i;$

$(ii)$  $\vec{r}=(g(u^2,\ldots,u^n)+u^1)\vec{a}+\int v(u^1) \vec{\eta}(u^1)du^1+\sum\limits_{i=2}^n u^i\int\lambda_i(u^1)\vec{\eta}(u^1)du^1;$

$(iii)$ $\vec{r}=(g(u^2,\ldots,u^n)+u^1)\vec{\rho}(u^1)+\int (v(u^1) - u^1) \dfrac{d \vec{\rho}(u^1)}{d u^1}du^1+\sum\limits_{i=2}^n u^i\int\lambda_i(u^1)\dfrac{d \vec{\rho}(u^1)}{d u^1}du^1.$

### Author Biography

O. O. Shugailo, School of Mathematics and Computer Sciences V.N. Karazin Kharkiv National University Kharkiv, Ukraine

School of Mathematics and Computer Sciences
V.N. Karazin Kharkiv National University
Kharkiv, Ukraine

### References

Magid M., Vrancken L. Flat affine surfaces in $mathbb{R}^4$ with flat normal connection// Geometriae Dedicata. – 2000. – V.81. – P. 19–31.

Nomizu K., Pinkall U. On the geometry of affine immersions// Mathematische Zeitschrift. – 1987. – Bd.195. – P. 165–178.

Nomizu K., Sasaki T. Affine differential geometry. – Cambridge University Press, 1994. – 264 p.

Nomizu K., Vrancken L. A new equiaffine theory for surfaces in $mathbb{R}^4$// International J. Math. – 1993. – V.4. – P. 127–165.

Shugailo E.A. Parallel affine immersions ${M}^nrightarrow {mathbb{R}}^{n+2}$ with flat connection// Ukr. Math. J. – 2014. – V.65, №9. – P. 1426–1445.

Shugailo O.O. Affine submanifolds of rank two// Journal of Math. Physics, Analysis, Geometry. – 2013. – V.9, №2. – P. 227–238.

Shugailo O.O. On affine immersions with flat connections// Journal of Math. Physics, Analysis, Geometry. – 2012. – V.8, №1. – P. 90–105.

Published
2023-09-22
How to Cite
Shugailo, O. O. (2023). Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mapping. Matematychni Studii, 60(1), 99-112. https://doi.org/10.30970/ms.60.1.99-112
Issue
Section
Articles